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arXiv2000

Chaotic behaviour of nonlinear waves and solitons of perturbed Korteweg - de Vries equation

This paper considers properties of nonlinear waves and solitons of Korteweg-de Vries equation in the presence of external perturbation. For time-periodic hamiltonian perturbation the width of the stochastic layer is calculated. The conclusions about chaotic behaviour in long-period waves and solitons are inferred. Obtained theoretical results find experimental confirmation in experiments with the propagation of ion-acoustic waves in plasma.

patt-sol↗
arXiv1998

Data-Adaptive Wavelets and Multi-Scale Singular Spectrum Analysis

Using multi-scale ideas from wavelet analysis, we extend singular-spectrum analysis (SSA) to the study of nonstationary time series of length $N$ whose intermittency can give rise to the divergence of their variance. SSA relies on the construction of the lag-covariance matrix C on M lagged copies of the time series over a fixed window width W to detect the regular part of the variability in that window in terms of the minimal number of oscillatory components; here W = M Dt, with Dt the time step. The proposed multi-scale SSA is a local SSA analysis within a moving window of width M <= W <= N. Multi-scale SSA varies W, while keeping a fixed W/M ratio, and uses the eigenvectors of the corresponding lag-covariance matrix C_M as a data-adaptive wavelets; successive eigenvectors of C_M correspond approximately to successive derivatives of the first mother wavelet in standard wavelet analysis. Multi-scale SSA thus solves objectively the delicate problem of optimizing the analyzing wavelet in the time-frequency domain, by a suitable localization of the signal's covariance matrix. We present several examples of application to synthetic signals with fractal or power-law behavior which mimic selected features of certain climatic and geophysical time series. A real application is to the Southern Oscillation index (SOI) monthly values for 1933-1996. Our methodology highlights an abrupt periodicity shift in the SOI near 1960. This abrupt shift between 4 and 3 years supports the Devil's staircase scenario for the El Nino/Southern Oscillation phenomenon.

chao-dyn↗
arXiv1999

Large deviations from the thermodynamic limit in globally coupled maps

Systems of a large number N of globally coupled maps have become popular as a relatively simple prototype of high-dimensional dynamics, showing many interesting and typical phenomena like synchronisation, cluster formation and multistability, and having potential applications in systems like Josephson junction arrays or in biophysical models. There exists a wealth of numerical investigations of globally coupled maps. While much progress has been made in the explanation of the macroscopic behaviour of such systems in the limit of infinite N, there is still need for a sound theory about the asymptotic behaviour of finite-N systems as N approaches infinity. This article introduces a method by which it is possible to obtain asymptotic estimates for long-term deviations from the thermodynamic limit behaviour. This method is based upon the concept of quasipotentials, originally developed by Freidlin, Wentzell, and others for describing the influence of small random perturbations on the long-term behaviour of dynamical systems. The problems of explicitly computing quasipotentials in the present context and potential approximation schemes are discussed. All the concepts described in this article are illustrated with a simple example.

physics.data-an↗
arXiv2000

On the dynamics of a self-gravitating medium with random and non-random initial conditions

The dynamics of a one-dimensional self-gravitating medium, with initial density almost uniform is studied. Numerical experiments are performed with ordered and with Gaussian random initial conditions. The phase space portraits are shown to be qualitatively similar to shock waves, in particular with initial conditions of Brownian type. The PDF of the mass distribution is investigated.

chao-dyn↗
arXiv2000

The rate of entropy increase at the edge of chaos

Under certain conditions, the rate of increase of the statistical entropy of a simple, fully chaotic, conservative system is known to be given by a single number, characteristic of this system, the Kolmogorov-Sinai entropy rate. This connection is here generalized to a simple dissipative system, the logistic map, and especially to the chaos threshold of the latter, the edge of chaos. It is found that, in the edge-of-chaos case, the usual Boltzmann-Gibbs-Shannon entropy is not appropriate. Instead, the non-extensive entropy $S_q\equiv \frac{1-\sum_{i=1}^W p_i^q}{q-1}$, must be used. The latter contains a parameter q, the entropic index which must be given a special value $q^*\ne 1$ (for q=1 one recovers the usual entropy) characteristic of the edge-of-chaos under consideration. The same q^* enters also in the description of the sensitivity to initial conditions, as well as in that of the multifractal spectrum of the attractor.

cond-mat.stat-mech↗
arXiv1998

A hierarchical scheme for cooperativity and folding in proteins

We propose a protein model based on a hierarchy of constraints that force the protein to follow certain pathways when changing conformation. The model exhibits a first order phase transition, cooperativity and is exactly solvable. It also shows an unexpected symmetry between folding and unfolding pathways as suggested by a recent experiment.

cond-mat↗
arXiv1997

Computing the Scaling Exponents in Fluid Turbulence from First Principles: the Formal Setup

We propose a scheme for the calculation from the NS equations of the scaling exponents $ζ_n$ of the $n$th order correlators in fully developed hydrodynamic turbulence. The scheme is nonperturbative and constructed to respect the fundamental rescaling symmetry of the Euler equation. It constitutes an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of $ζ_n$. As a consequence $ζ_n$ are determined by solvability conditions and not from power counting. It is arguedthat in order to achieve such a formulation one must recognize that the many-point space-time correlators are not scale invariant in time arguments. The assumption of full scale invariance leads unavoidably to K41 exponents. It is argued the determination of of $ζ_n$ requires equations for infinitely many renormalized objects. One ca n however proceed in controlled successive approximations by truncations of the infinite hierarchy of equations. Clues as to how to truncate without reintroducing power counting can be obtained from renormalized perturbation theory. To this aim we show that the fully resummed perturbation theory is equivalent in its contents to the exact hierarchy of equations obeyed by the $n$th order correlators and Green's function. In light of this important result we can safely use finite resummations to construct successive closures. This paper presents the conceptual and technical details of the scheme. The analysis of the high-order closure procedures which do not destroy the rescaling symmetry and the actual calculations for truncated models will be presented in a forthcoming paper in collaboration with V. Belinicher.

chao-dyn↗
arXiv1998

Fractional Calculus and the Evolution of Fractal Phenomena

It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we demonstrate that the fractional derivative (integral) of a generalized Weierstrass function (GWF) is another fractal function with a greater (lesser) fractal dimension. We also determine that the GWF is a solution to such a fractional differential stochastic equation of motion.

chao-dyn↗
arXiv1999

The existence of Burnett coefficients in the periodic Lorentz gas

The linear super-Burnett coefficient gives corrections to the diffusion equation in the form of higher derivatives of the density. Like the diffusion coefficient, it can be expressed in terms of integrals of correlation functions, but involving four different times. The power law decay of correlations in real gases (with many moving particles) and the random Lorentz gas (with one moving particle and fixed scatterers) are expected to cause the super-Burnett coefficient to diverge in most cases. Here we show that the expression for the super-Burnett coefficient of the periodic Lorentz gas converges as a result of exponential decay of correlations and a nontrivial cancellation of divergent contributions.

chao-dyn↗
arXiv1999

Robust and Irreversible Development in Cell Society as a General Consequence of Intra-Inter Dynamics

A dynamical systems scenario for developmental cell biology is proposed, based on numerical studies of a system with interacting units with internal dynamics and reproduction. Diversification, formation of discrete and recursive types, and rules for differentiation are found as a natural consequence of such a system. "Stem cells" that either proliferate or differentiate to different types stochastically are found to appear when intra-cellular dynamics are chaotic. Robustness of the developmental process against microscopic and macroscopic perturbations is shown to be a natural consequence of such intra-inter dynamics, while irreversibility in developmental process is discussed in terms of the gain of stability, loss of diversity and chaotic instability.

chao-dyn↗
arXiv1999

Chaotic dynamics and superdiffusion in a Hamiltonian system with many degrees of freedom

We discuss recent results obtained for the Hamiltonian Mean Field model. The model describes a system of N fully-coupled particles in one dimension and shows a second-order phase transition from a clustered phase to a homogeneous one when the energy is increased. Strong chaos is found in correspondence to the critical point on top of a weak chaotic regime which characterizes the motion at low energies. For a small region around the critical point, we find anomalous (enhanced) diffusion and Lévy walks in a transient temporal regime before the system relaxes to equilibrium.

cond-mat.stat-mech↗
arXiv1997

Universal spectral properties of spatially periodic quantum systems with chaotic classical dynamics

We consider a quasi one-dimensional chain of N chaotic scattering elements with periodic boundary conditions. The classical dynamics of this system is dominated by diffusion. The quantum theory, on the other hand, depends crucially on whether the chain is disordered or invariant under lattice translations. In the disordered case, the spectrum is dominated by Anderson localization whereas in the periodic case, the spectrum is arranged in bands. We investigate the special features in the spectral statistics for a periodic chain. For finite N, we define spectral form factors involving correlations both for identical and non-identical Bloch numbers. The short-time regime is treated within the semiclassical approximation, where the spectral form factor can be expressed in terms of a coarse-grained classical propagator which obeys a diffusion equation with periodic boundary conditions. In the long-time regime, the form factor decays algebraically towards an asymptotic constant. In the limit $N\to\infty$, we derive a universal scaling function for the form factor. The theory is supported by numerical results for quasi one-dimensional periodic chains of coupled Sinai billiards.

chao-dyn↗
arXiv1999

Analog to Digital Conversion in Physical Measurements

There exist measuring devices where an analog input is converted into a digital output. Such converters can have a nonlinear internal dynamics. We show how measurements with such converting devices can be understood using concepts from symbolic dynamics. Our approach is based on a nonlinear one-to-one mapping between the analog input and the digital output of the device. We analyze the Bernoulli shift and the tent map which are realized in specific analog/digital converters. Furthermore, we discuss the sources of errors that are inevitable in physical realizations of such systems and suggest methods for error reduction.

chao-dyn↗
arXiv2002

Transition to Turbulence in Coupled Maps on Hierarchical Lattices

General hierarchical lattices of coupled maps are considered as dynamical systems. These models may describe many processes occurring in heterogeneous media with tree-like structures. The transition to turbulence via spatiotemporal intermittency is investigated for these geometries. Critical exponents associated to the onset of turbulence are calculated as functions of the parameters of the systems. No evidence of nontrivial collective behavior is observed in the global quantity used to characterize the spatiotemporal dynamics.

chao-dyn↗
arXiv1998

Dissipation statistics of a passive scalar in a multidimensional smooth flow

We compute analytically the probability distribution function ${\cal P}(ε)$ of the dissipation field $ε=(\nabla θ)^{2}$ of a passive scalar $θ$ advected by a $d$-dimensional random flow, in the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime). The tail of the distribution is a stretched exponential: for $ε\to \infty$, $\ln {\cal P}(ε)\sim -(d^2ε)^{1/3}$.

chao-dyn↗
arXiv1999

A semiclassical theory of dissipative Henon-Heiles system

A semiclassical theory of dissipative Henon-Heiles system is proposed. Based on $\hbar$-scaling of an equation for evolution of Wigner quasiprobability distribution function in presence of dissipation and thermal diffusion, we derive a semiclassical equation for quantum fluctuations, governed by the dissipation and the curvature of the classical potential. We show how the initial quantum noise gets amplified by classical chaotic diffusion which is expressible in terms of correlation of stochastic fluctuations of the curvature of the potential due to classical chaos and ultimately settles down to equilibrium under the influence of dissipation. We also establish that there exists a critical limit to the expansion of phase space. The limit is set by chaotic diffusion and dissipation. Our semiclassical analysis is corroborated by numerical simulation of quantum operator master equation.

chao-dyn↗
arXiv1999

Entropy Production : From Open Volume Preserving to Dissipative Systems

We generalize Gaspard's method for computing the ε-entropy production rate in Hamiltonian systems to dissipative systems with attractors considered earlier by Tél, Vollmer, and Breymann. This approach leads to a natural definition of a coarse grained Gibbs entropy which is extensive, and which can be expressed in terms of the SRB measures and volumes of the coarse graining sets which cover the attractor. One can also study the entropy and entropy production as functions of the degree of resolution of the coarse graining process, and examine the limit as the coarse graining size approaches zero. We show that this definition of the Gibbs entropy leads to a positive rate of irreversible entropy production for reversible dissipative systems. We apply the method to the case of a two dimensional map, based upon a model considered by Vollmer, Tél and Breymann, that is a deterministic version of a biased-random walk. We treat both volume preserving and dissipative versions of the basic map, and make a comparison between the two cases. We discuss the ε-entropy production rate as a function of the size of the coarse graining cells for these biased-random walks and, for an open system with flux boundary conditions, show regions of exponential growth and decay of the rate of entropy production as the size of the cells decreases. This work describes in some detail the relation between the results of Gaspard, those of Tél, Vollmer and Breymann, and those of Ruelle, on entropy production in various systems described by Anosov or Anosov-like maps.

chao-dyn↗